### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 112, 6 pages      arXiv:1407.8291      https://doi.org/10.3842/SIGMA.2014.112

### Configurations of Points and the Symplectic Berry-Robbins Problem

Joseph Malkoun
Department of Mathematics and Statistics, Notre Dame University-Louaize, Lebanon

Received August 23, 2014, in final form December 17, 2014; Published online December 19, 2014

Abstract
We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group $\operatorname{Sp}(n)$, instead of the Lie group $\operatorname{U}(n)$. Denote by $\mathfrak{h}$ a Cartan algebra of $\operatorname{Sp}(n)$, and $\Delta$ the union of the zero sets of the roots of $\operatorname{Sp}(n)$ tensored with $\mathbb{R}^3$, each being a map from $\mathfrak{h} \otimes \mathbb{R}^3 \to \mathbb{R}^3$. We wish to construct a map $(\mathfrak{h} \otimes \mathbb{R}^3) \backslash \Delta \to \operatorname{Sp}(n)/T^n$ which is equivariant under the action of the Weyl group $W_n$ of $\operatorname{Sp}(n)$ (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of $\operatorname{Sp}(n)$, and $T^n$ is the diagonal $n$-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for $n=2$.

Key words: configurations of points; symplectic; Berry-Robbins problem; equivariant map; Atiyah-Sutcliffe problem.

pdf (251 kb)   tex (10 kb)

References

1. Atiyah M., The geometry of classical particles, in Surveys in Differential Geometry, Surv. Differ. Geom., Vol. 7, Int. Press, Somerville, MA, 2000, 1-15.
2. Atiyah M., Configurations of points, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), 1375-1387.
3. Atiyah M., Bielawski R., Nahm's equations, configuration spaces and flag manifolds, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 157-176, math.RT/0110112.
4. Atiyah M., Sutcliffe P., The geometry of point particles, Proc. Roy. Soc. London Ser. A 458 (2002), 1089-1115, hep-th/0105179.
5. Berry M.V., Robbins J.M., Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proc. Roy. Soc. London Ser. A 453 (1997), 1771-1790.